When I first heard of Kimberlé Crenshaw's theory of intersectionality, I loved it because I thought it explained the way that different prejudices interact with different people. I misunderstood it entirely.
Intersectionality assumes all prejudices are unique. Identitarians love it because it was founded on the notion that different forms of oppression have no connections; they only have intersections. It is extremely convenient for capitalists who don't want you to wonder about underlying causes.
Its great problem is its inherent anti-scientific nature—it rules out the possibility of interrelatedness. It leads social justice warriors like Coffeeandink to say things like, "...I do think class is a significant axis of oppression separate from but interacting with race and gender. I just don't think it's the root oppression that is the basis of all other oppression, or that eliminating class injustice will magically cause other forms of prejudice and injustice to fade away."
Now, the last clause is a common straw man, but as for the basic identitarian take on class, race and gender being separate—
Historians like the eminent Trinidadan historian Eric Williams disagree: “Slavery was not born of racism: rather, racism was the consequence of slavery.” People who use "race" and "racism" in wildly imprecise ways have trouble understanding what he meant, but until the development of the African slave trade, prejudice was tribal, not racial—perhaps the most famous example is the Greek prejudice against barbarians, people whose talk sounded like "bar-bar-bar" to them.
Socialist feminists also disagree with intersectionality. Engels wrote, "The first class opposition that appears in history coincides with the development of the antagonism between man and woman in monogamous marriage, and the first class oppression coincides with that of the female sex by the male." While Engels is not saying that sexual oppression is class oppression, he's suggesting a link that later writers would explore about the relationship between the nature of economic hierarchy—what middle-class feminists call kyriarchy—and sexual exploitation.
While "kyriarchy" strikes me as a redundant word, it is fascinating that capitalist feminists are fumbling for a broader understanding of oppression. It may force them to abandon intersectionality.
Recommended:
RACE - The Power of an Illusion . Go Deeper | PBS.
Slavery and the origins of racism by Lance Selfa.
'The Whiting of Euro-Americans: A Divide and Conquer Strategy', by Rev. Thandeka.
Marxist Internet Archive Library of Feminist Writers.
ETA: After considering the discussion in the comments, I wrote do different forms of oppression intersect or dogpile?